Surface Area of Sphere

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Theorem

The surface area $A$ of a sphere whose radius $a$ is given by:

$A = 4 \pi a^2$


Proof


Historical Note

The Surface Area of Sphere was demonstrated by Archimedes

His proof appears as Proposition $34$ in his On the Sphere and Cylinder.


He also discusses this result in his The Method:

From this theorem, to the effect that a sphere is four times as great as the cone with a great circle of the sphere as base and height equal to the radius of the sphere, I conceived the notion that the surface of any sphere is four times as great as a great circle in it; for, judging from the fact that any circle is equal to a triangle with base equal to the circumference and height equal to the radius of the circle, I apprehended that, in like manner, a sphere is equal to a cone with base equal to the surface of the sphere and height equal to the radius.